Triangulated categories and derived categories were introduced by
Grothendieck and Verdier [1]}. Today, they have widely been
used in many branches: algebraic geometry, stable homotopy theory,
representation theory, etc. In the representation theory of
algebras, we will restrict our attention to the equivalences of
derived categories, that is, derived equivalences. Derived
equivalences have been shown to preserve many invariants and provide
new connection. For instance, Hochschild homology and cohomology
[2]}, finiteness of finitistic dimension [3]} have been
shown to be invariant under derived equivalences. Moreover, derived
equivalences are related to cluster categories and cluster tilting
objects [4]}. As is known, Rickard's Morita theory for
derived categories leaves something to be desired, though, as for
some pairs of rings, or algebras, it is currently difficult,
sometimes even impossible to verify whether there exists a tilting
complex. It is of interest to construct a new derived equivalence
from given one by finding a suitable tilting complex. Rickard
[5]}, [2]} used tensor products and trivial extensions to get
new derived equivalences. In the recent years, Hu and Xi have
provided various techniques to construct new derived equivalences.
In [7]} they established an amazing connection between derived
equivalences and Auslander-Reiten sequences via BB-tilting modules,
and obtained derived equivalences from Auslander-Reiten triangles.
In [8]} they constructed new derived equivalences between
\(\Phi \) -Auslander-Yoneda algebras from a given almost \(\nu \) -stable
equivalence.
